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{n}{0}+2{n}{1}+2^2{n}{2}+ .... +2^n{n}{n} is equal to

Question

$(0 n) + 2 (1 n) + 2^{2} (2 n) + .... + 2^{n} (n n)$ is equal to

2^{n}

0

3^{n}

none of these

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Sebastian Discussed

(0 n) + 2 (1 n) + 2^{2} (2 n) + .... + 2^{n} (n n)

is equal to

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Text solutionVerified

Given,

(0 n) + 2 (1 n) + 2^{2} (2 n) + .... + 2^{n} (n n)

Simplifying the above expression we get

=^{n} C_{0} 2^{0} +^{n} C_{1} 2^{1} +^{n} C_{2} 2^{2} + ...^{n} C_{n} 2^{n}

Consider

x = 2

in the expansion of

(1 + x)^{n}

= (1 + 2)^{n}

= 3^{n}

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